# 0.3 Signal processing in processing: sampling and quantization  (Page 3/4)

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## 2-d: images

Let us assume we have a continuous distribution, on a plane, of values of luminance or, more simply stated, animage. In order to process it using a computer we have to reduce it to a sequence of numbers by means ofsampling. There are several ways to sample an image, or read its values of luminance at discrete points. Thesimplest way is to use a regular grid, with spatial steps $X$ e $Y$ . Similarly to what we did for sounds, we define the spatial sampling rates ${F}_{X}=\frac{1}{X}$ and ${F}_{Y}=\frac{1}{Y}$ . As in the one-dimensional case, also for two-dimensional signals, or images, sampling can bedescribed by three facts and a theorem.

• The Fourier Transform of a discrete-space signal is a function (called spectrum ) of two continuous variables ${\omega }_{X}$ and ${\omega }_{Y}$ , and it is periodic in two dimensions with periods $2\pi$ . Given a couple of values ${\omega }_{X}$ and ${\omega }_{Y}$ , the Fourier transform gives back a complex number that can be interpreted as magnitude andphase (translation in space) of the sinusoidal component at such spatial frequencies.
• Sampling the continuous-space signal $s(x, y)$ with the regular grid of steps $X$ , $Y$ , gives a discrete-space signal $s(m, n)=s(mX, nY)$ , which is a function of the discrete variables $m$ and $n$ .
• Sampling a continuous-space signal with spatial frequencies ${F}_{X}$ and ${F}_{Y}$ gives a discrete-space signal whose spectrum is the periodic replication along the grid of steps ${F}_{X}$ and ${F}_{Y}$ of the original signal spectrum. The Fourier variables ${\omega }_{X}$ and ${\omega }_{Y}$ correspond to the frequencies (in cycles per meter) represented by the variables ${f}_{X}=\frac{{\omega }_{X}}{2\pi X}$ and ${f}_{Y}=\frac{{\omega }_{Y}}{2\pi Y}$ .

The [link] shows an example of spectrum of a two-dimensional sampled signal. There, thecontinuous-space signal had all and only the frequency components included in the central hexagon. The hexagonalshape of the spectral support (region of non-null spectral energy) is merely illustrative. The replicas of the originalspectrum are often called spectral images .

Given the above facts , we can have an intuitive understanding of the Sampling Theorem.

## Sampling theorem (in 2d)

A continuous-space signal $s(x, y)$ , whose spectral content is limited to spatial frequencies belonging to the rectangle having semi-edges ${F}_{bX}$ and ${F}_{bY}$ (i.e., bandlimited) can be recovered from its sampled version $s(m, n)$ if the spatial sampling rates are larger than twice the respective bandwidths (i.e., if ${F}_{X}> 2{F}_{bX}$ and ${F}_{Y}> 2{F}_{bY}$ )

In practice, the spatial sampling step can not be larger than the semi-period of the finest spatial frequency (or thefinest detail) that is represented in the image. The reconstruction can only be done through a filter thateliminates all the spectral images but the one coming directly from the original continuous-space signal. In otherwords, the filter will cut all images whose frequency components are higher than the Nyquist frequency defined as $\frac{{F}_{X}}{2}$ and $\frac{{F}_{Y}}{2}$ along the two axes. The condition required by the sampling theorem is equivalent to requiring that there are no overlaps between spectral images. If there were suchoverlaps, it wouldn't be possible to eliminate the copies of the original signal spectrum by means of filtering. In case of overlapping, a filtercutting all frequency components higher than the Nyquist frequency would give back a signal that is affected byaliasing.

#### Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
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Daniel
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Abigail
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Anassong
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NANO
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are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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Tarell
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Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
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Cied
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what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
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Porter
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